We provide an efficient data structure for dynamically maintaining a
ham-sandwich cut of two non-overlapping convex polygons in the plane.
Given two non-overlapping convex polygons P1 and
P2 in the plane,
the ham-sandwich cut of P1 and P2
is a line that
simultaneously bisects the area (or perimeter or vertex count)
of both polygons. We provide a data structure that
supports queries for the ham-sandwich cut in
O(log3n) worst-case time and
insertions and deletions of vertices of the
Pi in O(log n)
worst-case time.
We also show how this data structure can be used to maintain a
partition of the plane by two lines into four regions each containing a
quarter of the total polygon area (or perimeter or vertex count).
In particular, if we use the vertex-count measure, the intersection of these
two lines gives a point of Tukey depth n/4,
which serves as an approximate Tukey median.